Advertisement

Optimization Letters

, Volume 12, Issue 2, pp 399–410 | Cite as

A weighting subgradient algorithm for multiobjective optimization

  • G. C. Bento
  • J. X. Cruz Neto
  • P. S. M. Santos
  • S. S. Souza
Original Paper
  • 209 Downloads

Abstract

We propose a weighting subgradient algorithm for solving multiobjective minimization problems on a nonempty closed convex subset of an Euclidean space. This method combines weighting technique and the classical projected subgradient method, using a divergent series steplength rule. Under the assumption of convexity, we show that the sequence generated by this method converges to a Pareto optimal point of the problem. Some numerical results are presented.

Keywords

Pareto optimality Multiobjective optimization Projected subgradient method Weighting method 

Notes

Acknowledgements

First author was supported in part by CAPES-MES-CUBA 226/2012, FAPEG 201210267000909-05/2012 and CNPq Grants 458479/2014-4, 471815/2012-8, 303732/2011-3, 312077/2014-9. The second was partially supported by CNPq Grant 305462/2014-8 and PRONEX Optimization(FAPERJ/CNPq). The third was supported in part by CNPq Grant 485205/2013-0 and PROPESQ/UFPI.

References

  1. 1.
    Alber, Y.I., Iusem, A.N., Solodov, M.V.: On the projected subgradient method for nonsmooth convex optimization in a Hilbert space. Math. Program. 81, 23–35 (1998)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bello Cruz, J.Y.: A subgradient method for vector optimization problems. SIAM J. Optim. 23(4), 2169–2182 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bento, G.C., Cruz Neto, J.X.: A subgradient method for multiobjective optimization on Riemannian manifolds. J. Optim. Theory Appl. 159, 125–137 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brito, A.S., Cruz Neto, J.X., Santos, P.S.M., Souza, S.S.: A relaxed projection method for solving multiobjective optimization problems. Eur. J. Oper. Res. 256, 17–23 (2016). doi: 10.1016/j.ejor.2016.05.026
  5. 5.
    Burachik, R.S., Kaya, C.Y., Rizvi, M.M.: A new scalarization technique to approximate Pareto fronts of problems with disconnected feasible sets. J. Optim. Theory Appl. 162, 428–446 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fliege, J., Graña Drummond, L.M., Svaiter, B.F.: Newton’s method for multiobjective optimization. SIAM J. Optim. 20(2), 602–626 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51, 479–494 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fukuda, E.H., Graña Drummond, L.M.: Inexact projected gradient method for vector optimization. Comput. Optim. Appl. 54(3), 473–493 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Graña Drummond, L.M., Maculan, N., Svaiter, B.F.: On the choice of parameters for the weighting method in vector optimization. Math. Program. 111, 201–216 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Huband, S., Hingston, P., Barone, L., While, L.: A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans. Evol. Comput. 10(5), 477–506 (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989)Google Scholar
  12. 12.
    Marler, R.T., Arora, S.J.: The weighted sum method for multiobjective optimization: new insights. Struct. Multidiscip. Optim. 41(6), 853–862 (2009)CrossRefzbMATHGoogle Scholar
  13. 13.
    Miettinen, K.M.: Nonlinear Multiobjective Optimization. Kluwer Academic, Norwell (1999)zbMATHGoogle Scholar
  14. 14.
    Pappalardo, M.: Multiobjective optimization: a brief overview. In: Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds.) Pareto Optimality, Game Theory and Equilibria. Springer Optimization and Its Applications, vol. 17, pp. 517–528. Springer, New York (2008)Google Scholar
  15. 15.
    Pardalos, P.M., Steponaviče, I., Žilinskas, A.: Pareto set approximation by the method of adjustable weights and successive lexicographic goal programming. Optim. lett. 6, 665–678 (2012)Google Scholar
  16. 16.
    Polyak, B.T.: Introduction to Optimization. Translations Series in Mathematics and Engineering. Optimization Software, Inc., New York (1987)Google Scholar
  17. 17.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • G. C. Bento
    • 1
  • J. X. Cruz Neto
    • 2
  • P. S. M. Santos
    • 3
  • S. S. Souza
    • 3
  1. 1.IME/UFGGoianiaBrazil
  2. 2.DM/UFPITeresinaBrazil
  3. 3.CMRV/UFPIParnaíbaBrazil

Personalised recommendations